What is this G button function in blender called(happens in Edit mode)?
This is the Edge Slide function. Selecting an edge and pressing G two times will enable this. It allows you to manipulate edges along other edges, scaling then accordingly to maintain the overall shape while moving the edge. An example can be seen below on my mesh:



Hope this helps.

EDIT:
On your video, in the bottom left corner, you can see you have "Vert Slide" enabled. This works similar to Edge Slide.